lec13 - Theorem 13.3. Let ( t n ) be a sequence of real...

Math 117 – May 17, 2011 13 Convergence criteria for sequences Monotone Sequences Deﬁnition 13.1. A sequence of real numbers ( s n ) is called increasing , if s n +1 ≥ s n for all n ∈ N . The sequence is called decreasing , if s n +1 ≤ s n for all n ∈ N . A sequence is called monotone , if it is decreasing or increasing. A sequence would be called strictly increasing/decreasing, if the inequalities in the deﬁnition are strict. The ﬁrst criteria for convergence of a sequence uses monotonicity and boundedness. Theorem 13.2 (Monotone convergence criterion) . Let ( s n ) be a sequence of real numbers. ( a ) If ( s n ) is increasing and bounded above, then ( s n ) converges. ( b ) If ( s n ) is decreasing and bounded below, then ( s n ) converges. It is not hard to see what the behavior of a monotone unbounded sequence is.

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